Theta-summable spectral triple - Imported Knowlpedia Content

Definition

A spectral triple $(\mathcal A,H,D)$ is theta-summable if

e^{-tD^2}

is trace class for every $t>0$ , equivalently if $\operatorname{Tr}(e^{-tD^2})<\infty$ for every $t>0$ , where $\operatorname{Tr}$ is the canonical operator trace. The exponential is defined by functional calculus for the self-adjoint operator $D$ . This condition controls the multiplicity and growth of the spectrum without requiring any fixed power of $(1+D^2)^{-1/2}$ to be trace class. Consequently, theta summability is weaker than finite $p$ -summability but strong enough to make heat-kernel cochains, notably the Jaffe–Leśniewski–Osterwalder cochain, analytically meaningful.

Comparison with finite summability

If $(1+D^2)^{-p/2}$ is trace class for some $p>0$ , then the triple is theta-summable: exponential decay dominates every negative power. The converse fails. For example, an eigenvalue counting function may grow faster than every polynomial while remaining subexponential in the sense required for $\operatorname{Tr}(e^{-tD^2})$ to converge at every positive time.

Role in entire cyclic cohomology

Products containing heat factors $e^{-t_jD^2}$ regularize the unbounded commutators that occur in the JLO formula. Integration over a simplex and trace-norm estimates then give the factorial growth bounds required of an entire cyclic cochain. This is the original analytic setting of Jaffe–Leśniewski–Osterwalder, §§2–4.

Conventions and scope

References

A. Jaffe, A. Leśniewski, and K. Osterwalder, “Quantum K-Theory. I. The Chern Character,” Communications in Mathematical Physics 118 (1988), 1–14. DOI record. Relevant: theta-summability and the entire Chern character.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: chapter IV on theta-summable Fredholm modules and entire cyclic cohomology.